Concept explainerCalculus IFoundational7 min read

What continuity at a point really requires

Three conditions, one precise promise: nearby inputs produce nearby outputs without a break at the point.

\lim_{x\to a}f(x)=f(a)

Concept explainer

What the idea means, why its conditions matter, and where it connects.

Reviewed July 11, 2026

DefinitionStart here

\boxed{\lim_{x\to a}f(x)=f(a)}

The function must be defined at a, its two-sided limit must exist, and that limit must equal the assigned value.

The three-condition checklist

First, f(a) must exist. Second, the left- and right-hand limits must agree. Third, the shared limit must equal f(a). Missing any one condition produces a discontinuity.

This separates holes, jumps, and vertical blowups instead of calling every visual break the same thing.

Continuity is local

A function can be continuous at one point and fail nearby. The definition makes a claim only about inputs sufficiently close to a and the value at a itself.

For familiar elementary functions, continuity lets you evaluate limits by substitution wherever the expression is defined.

Repairing a removable discontinuity

If the limit exists but the function value is missing or wrong, assigning the limiting value repairs the point. This changes one output without changing the nearby rule.

Worked exampleChoose the value that closes the hole

Define f(2) so that (x² − 4)/(x − 2) becomes continuous at x = 2.

\frac{x^2-4}{x-2}=x+2\quad(x\ne2)

Factor and simplify the nearby behavior.

\lim_{x\to2}f(x)=4

The limit exists even before the missing value is assigned.

f(2)=4

Match the point value to the limit.

Result

\boxed{f(2)=4}

Watch for

Common mistakes

Checking only that f(a) exists.
Assuming a graph is continuous because the pieces touch visually.
Using a two-sided limit when the domain naturally ends at the point.

Keep

Three takeaways

Defined, limit exists, and values match.
Continuity turns many limits into substitution.
A removable hole can be repaired with one value.